Fig. 1: Experimental platform and theoretical framework.

a, A delta-doped layer of ¹⁴N (green) is grown on a diamond substrate. NV centres are created via local electron irradiation (orange beam) and subsequent high-temperature annealing. b, Schematic depiction of a two-dimensional layer of NV (red) and P1 (blue) centres. Dilute NV centres function as probe spins of the dense, disordered P1 system. The P1 centres exhibit spin-flip dynamics driven by magnetic dipole–dipole interactions (zoom). Ising interactions with the P1 system cause the NV to accumulate a phase, ϕ, during noise spectroscopy (Bloch sphere). c, NV and P1 level structure in the presence of a magnetic field, B, applied along the NV axis. We work within an effective spin 1/2 subspace of the NV centre, \(\{\left\vert 0\right\rangle ,\left\vert -1\right\rangle \}\), with level splitting, ω_NV. The corresponding P1 splitting, ω_P1, is strongly off-resonant from the NV transition. d, Secondary ion mass spectrometry measurement of the density of ¹⁴N as a function of depth for sample S1. The presence of a thin layer is indicated by a sharp nitrogen peak with a 8-nm width, limited by the secondary ion mass spectrometry resolution. e, The overlap between the many-body spectral function (blue) and the power spectrum of the filter function ∣f(ω; t)∣² determines the variance of the phase ∼ χ(t) (equation (2)). ∣f(ω; t)∣² for both a Ramsey/DEER pulse sequence (purple) and a spin echo pulse sequence (orange) are shown. f, Schematic depiction of the variance of the phase, \(\left\langle {\phi }^{2}\right\rangle =-2\log C(t)\), as a function of the measurement duration t, for both Ramsey/DEER (purple) and spin echo (orange). The labelled slopes indicate the predicted stretch powers in both the early-time ballistic regime and the late-time random-walk regime (Table 1). The cross-over occurs at the correlation time, τ_c.

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